chapter 15 The Differential Calculus

THE DIFFERENTIAL CALCULUS

THE INVENTION of the differential calculus marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level. These contrasted periods in the progress of the history of thought are compared by Shelley to the formation of an avalanche.

微积分的发明标志着数学史上的一次变革。科学的进步可以分为两种时期：一种是思想的缓慢积累期，另一种则是由于前者耐心收集的思想材料，一位天才通过发明一种新方法或提出新的观点，突然将整个学科提升到更高的层次。这种思想史上对比鲜明的时期，被雪莱比作雪崩的形成过程。 The sun-awakened avalanche! whose mass,

Thrice sifted by the storm, had gathered there

Flake after flake,——in heaven-defying minds

As thought by thought is piled, till some great truth

Is loosened, and the nations echo rounds, 太阳唤醒的雪崩！其巨大的雪团，

被风暴三次筛选，在那里汇聚，

一片又一片，——如同在蔑视天堂的思想者心中，

思想一点一点堆积，直到某个伟大的真理

被释放，万国回响其声。

The comparison will bear some pressing. The final burst of sunshine which awakens the avalanche is not necessarily beyond comparison in magnitude with the other powers of nature which have presided over its slow formation. The same is true in science. The genius who has the good fortune to produce the final idea which transforms a whole region of thought, does not necessarily excel all his predecessors who have worked at the history of science, it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances towards a new epoch.

这个比喻值得深入探讨。唤醒雪崩的最后一缕阳光，其规模未必远远超过在其缓慢形成过程中起作用的其他自然力量。科学领域也是如此。那个有幸提出最终思想，从而改变整个思想领域的天才，并不一定在才能上超越所有曾为科学史做出贡献的前人。只对那些推动新纪元最后一步的人怀着惊讶的崇拜，而忽视之前所有的努力，这既愚蠢，又忘恩负义。

In the particular instance before us, the subject had a long history before it assumed its final form at the hands of its two inventors. There are some traces of its methods even among the Greek mathematicians, and finally, just before the actual production of the subject, Fermat (born A.D. 1601 and died A.D. 1665), a distinguished French mathematician, had so improved on previous ideas that the subject was all but created by him. Fermat, also, may lay claim to be the joint inventor of co-ordinate geometry in company with his contemporary and countryman, Descartes. It was, in fact, Descartes from whom the world of science received the new ideas, but Fermat had certainly arrived at them independently.

在我们面前的这个特定案例中，这一学科在最终由两位发明者确立其最终形式之前，已有着悠久的历史。甚至在希腊数学家中，也可以找到其方法的一些痕迹。最终，在这一学科正式诞生之前，法国杰出数学家费马（生于公元1601年，卒于公元1665年）对先前的思想进行了极大的改进，以至于这一学科几乎完全由他创建。此外，费马也可以声称自己与他的同时代人、同胞笛卡尔共同发明了坐标几何。事实上，科学界是通过笛卡尔才得以接受这些新思想的，但费马无疑是独立得出这些思想的。

We need not, however, stint our admiration either for Newton or for Leibniz. Newton was a mathematician and a student of physical science, Leibniz was a mathematician and a philosopher, and each of them in his own department of thought was one of the greatest men of genius that the world has known. The joint invention was the occasion of an unfortunate and not very creditable dispute. Newton was using the methods of Fluxions, as he called the subject, in 1666, and employed it in the composition of his Principia, although in the work as printed any special algebraic notation is avoided. But he did not print a direct statement of his method till 1693. Leibniz published his first statement in 1684. He was accused by Newton's friends of having got it from a manuscript by Newton, which he had been shown privately. Leibniz also accused Newton of having plagiarized from him. There is now very much doubt but that both should have the credit of being independent discoverers. The subject had arrived at a stage in which it was ripe for discovery, and there is nothing surprising in the fact that two such able men should have independently hit upon it.

然而，我们不必因此吝惜对牛顿或莱布尼茨的赞赏。牛顿既是数学家，又是物理科学的研究者；莱布尼茨则既是数学家，又是哲学家。在各自的思想领域中，他们都是世界上最伟大的天才之一。这一共同发明引发了一场不幸且不太光彩的争端。牛顿在1666年就已经使用他称之为“流数法”（Fluxions）的方法，并在撰写 《自然哲学的数学原理》 (Principia) 时运用了这一方法，尽管该书的正式出版版本中刻意避免了任何特殊的代数符号。然而，他直到1693年才直接发表了对该方法的说明。而莱布尼茨则在1684年首次发表了自己的研究成果。牛顿的支持者指控莱布尼茨从牛顿的手稿中窃取了这一发现，因为莱布尼茨曾私下看到过牛顿的手稿。与此同时，莱布尼茨也反过来指责牛顿抄袭了自己的研究。如今，人们普遍认为，两人应当被视为独立的发现者。这一学科当时已发展到成熟阶段，正处于被发现的临界点，因此，两位如此杰出的学者能够各自独立地得出相同的结论，并不令人意外。

These joint discoveries are quite common in science. Discoveries are not in general made before they have been led up to by previous trend of thought, and by that time many minds are in hot pursuit of the important idea. If we merely keep to discoveries in which Englishmen are concerned, the simultaneous enunciation of the law of natural selection by Darwin and Wallace, and the simultaneous discovery of Neptune by Adams and the French astronomer, Leverrier, at once occur to the mind. The disputes, as to whom the credit ought to be given, are often influenced by an unworthy spirit of nationalism. The really inspiring reflection suggested by the history of mathematics is the unity of thought and interest among men of so many epochs, so many nations, and so many races. Indians, Egyptians, Assyrians, Greeks, Arabs, Italians, Frenchmen, Germans, Englishmen, and Russians, have all made essential contributions to the progress of the science. Assuredly the jealous exaltation of the contribution of one particular nation is not to show the larger spirit.

这些共同的发现在科学领域相当常见。一般而言，重大发现不会在没有先前思想发展的铺垫下突然出现，而当这一时刻到来时，许多学者都在全力追寻这一重要的思想。如果我们仅限于讨论与英国人相关的发现，那么达尔文与华莱士对自然选择法则的同时提出，以及亚当斯与法国天文学家勒维耶对海王星的同时发现，立刻浮现在脑海中。围绕这些发现归功于何人而展开的争论，往往受到一种狭隘的民族主义情绪所影响。数学史带给我们的真正鼓舞人心的思考，是跨越不同时代、不同国家、不同种族的人们在思想和兴趣上的统一性。印度人、埃及人、亚述人、希腊人、阿拉伯人、意大利人、法国人、德国人、英国人和俄罗斯人，都为科学的进步做出了至关重要的贡献。毫无疑问，那种出于嫉妒心理而过分夸大某个国家贡献的做法，并不能体现更高远的精神境界。

The importance of the differential calculus arises from the very nature of the subject, which the systematic consideration of the rates of increase of functions. This idea is immediately presented to us by the study of nature; velocity is the rate of increase of the distance travelled, and acceleration is the rate of increase of velocity. Thus the fundamental idea of change, which is at the basis of our whole perception of phenomena, immediately suggests the inquiry as to the rate of change. The familiar terms of 'quickly' and 'slowly' gain their meaning from a tacit reference to rates of change. Thus the differential calculus is concerned with the very key of the position from which mathematics can be successfully applied to the explanation of the course of nature.

微积分的重要性源于其研究对象的本质，即对函数增长速率的系统性考察。这个概念直接来自于我们对自然的研究：速度是物体运动距离的变化率，加速度则是速度的变化率。因此，“变化”这一基本概念——它构成了我们对现象整体认知的基础——自然引发了人们对变化速率的探究。我们日常使用的“快”和“慢”这些词，其意义正是基于对变化速率的隐含参照。因此，微积分研究的正是数学能够成功应用于自然现象解释的核心关键。

This idea of the rate of change was certainly in Newton's mind, and was embodied in the language in which he explained the subject. It may be doubted, however, whether this point of view, derived from natural phenomena, was ever much in the minds of the preceding mathematicians who prepared the subject for its birth. They were concerned with the more abstract problems of drawing tangents to curves, of finding the lengths of curves, and of finding the areas enclosed by curves. The last two problems, of the rectification of curves and the quadrature of curves as they are named, belong to the Integral Calculus, which is, however, involved in the same general subject as the Differential Calculus.

变化率的概念无疑存在于牛顿的思维中，并体现在他解释这一学科所使用的语言中。然而，我们可以怀疑，这种从自然现象中获得的观点是否曾经在为微积分诞生做准备的前辈数学家的思维中占据重要位置。他们更关注的是一些更为抽象的问题，比如作曲线的切线、求曲线的长度以及计算曲线围成的面积。后两个问题，即曲线的求长（rectification of curves）和曲线的求面积（quadrature of curves），属于积分学（Integral Calculus）的范畴。然而，积分学与微分学（Differential Calculus）实际上属于同一数学体系。

image-20250305160431402

The introduction of co-ordinate geometry makes the two points of view coalesce. For (cf. Fig. 32) let be any curved line and let be the tangent at the point on it. Let the axes of co-ordinate be and ; and let be the equation to the curve, so that and . Now let be any moving point on the curve, with co-ordinates ; then . And let be the point on the tangent with the same abscissa ; suppose that the co-ordinates of are and . Now suppose that moves along the axis from left to right with a uniform velocity; then it is easy to see that the ordinate of the point on the tangent also increases uniformly as moves along the tangent in a corresponding way. In fact it is east to see that the ratio of the rate of increase of to the rate of increase of is in the ratio of to , which is the same at all points of the straight line. But the rate of increase of , which is the rate of increase of , varies from point to point of the curve so long as it is not straight. As passes through the point , the rate of increase of (where coincides with for the moment) is the same as the rate of increase of on the tangent at . Hence, if we have a general method of determining the rate of increase of function of a variable , we can determine the slope of the tangent at any point on a curve, and thence can draw it. Thus the problems of drawing tangents to a curve, and of determining the rate of increase of function are really identical.

解析几何的引入使这两种观点合二为一。

设 是任意一条曲线， 是其上点 处的切线（见图 32）。设坐标轴为 和 ，且曲线的方程为 ，则有 和 。

现在，设 是曲线上的任意移动点，其坐标为 ，因此 。令 为切线 上横坐标与 相同的点，假设其坐标为 。

现在，假设点 以恒定速度沿 轴从左向右移动，那么很容易看出，切线上点 的纵坐标 也会随之均匀增加。事实上，可以证明， 的增加速率与 的增加速率之比，等于 与 之比，而该比值在整条直线上是恒定的。

然而，曲线上的 的增长速率（即 的增长速率）在曲线各点处是不同的，只要曲线不是直线。当点 经过点 时，函数 的增长速率（此时 短暂地与 重合）与切线 上 的增长速率相同。因此，如果我们有一个通用的方法来确定函数 关于变量 的增长速率，那么我们就可以确定曲线任意一点 处切线的斜率，并据此作出切线。

由此可见，作曲线的切线问题与求函数的变化率问题本质上是相同的。

It will be noticed that , as in the cases of Conic Sections and Trigonometry, the more artificial of the two points of view is the one in which the subject took its rise. The really fundamental aspect of the science only rose into prominence comparatively late in the day. It is a well-founded historical generalization, that the last thing to be discovered in any science is what the science is really about. Men go on groping for centuries, guided merely by a dim instinct and a puzzled curiosity, till at last 'some great truth is loosened'.

可以注意到，就像圆锥曲线和三角学的情况一样，这两种观点中较为人为的一种，反而是该学科最初发展的出发点。而该科学真正基本的方面，直到相对较晚的时期才变得显著。这是一条有充分依据的历史概括：在任何科学中，最后被发现的，往往是该科学真正研究的核心内容。人们往往在黑暗中摸索了几个世纪，仅凭模糊的直觉和困惑的好奇心前行，直到最终“某个伟大的真理被揭示出来”。

Let us take some special cases in order to familiarize ourselves with the sort of ideas which we want to make precise. A train is in motion —— how shall we determine its velocity at some instant, let us say, at noon? We can take an interval of five minutes which includes noon, and measure how far the train has gone in that period. Suppose we find it to be five miles, we may then conclude that the train was running at the rate of 60 miles per hour. But five miles is a long distance, and we cannot be sure that just at noon the train was moving at this pace. At noon it may have been running 70 miles per hour, and afterwards the brake may have been put on. It will be safer to work with a smaller interval, say one minute, which includes noon, and to measure the space traversed during that period. But for some purposes greater accuracy may be required, and one minute may be too long. In practice, the necessary inaccuracy of our measurements makes it useless to take too small a period for measurement. But in theory the smaller the period the better, and we are tempted to say that for ideal accuracy an infinitely small period is required. The older mathematicians, in particular Leibniz, were not only tempted, but yielded to the temptation, and did say it. Even now it is a useful fashion of speech, provided that we know how to interpret it into the language of common sense. It is curious that , in his exposition of the foundations of the calculus, Newton, the natural scientist, is much more philosophical than Leibniz, the philosopher, and on the other hand, Leibniz provided the admirable notation which has been so essential for the progress of the subject.

让我们通过一些特殊情况来熟悉我们想要精确化的这类概念。一列火车正在行驶——我们该如何确定它在某一时刻（比如正午）的速度呢？我们可以取一个包含正午的五分钟时间间隔，并测量火车在这段时间内行驶的距离。假设我们发现它在这段时间内行驶了五英里，那么我们可以得出结论，火车的行驶速度为每小时 60 英里。然而，五英里是一个相对较长的距离，因此我们无法确定火车在正午时刻的确切速度。或许在正午时刻，火车的速度是每小时 70 英里，而之后才开始刹车。为了获得更精确的结果，我们可以选择一个更小的时间间隔，例如包含正午的一分钟，并测量火车在这段时间内的行驶距离。但是，在某些情况下，我们可能需要更高的精度，一分钟的时间间隔可能仍然太长。实际上，由于测量误差的存在，选择过小的时间间隔会变得毫无意义。但在理论上，时间间隔越小，结果就越精确，因此我们可能会认为，要达到理想的精度，需要一个无限小的时间间隔。早期的数学家们，尤其是莱布尼茨，不仅受到了这种想法的诱惑，而且直接接受了它，并明确地提出了这一观点。即使在今天，这种表述方式仍然很有用，前提是我们能够将其合理地解释为符合常识的语言。有趣的是，在对微积分基础的论述中，作为自然科学家的牛顿比哲学家莱布尼茨更具哲学思辨，而另一方面，莱布尼茨却提供了极为精妙的数学符号表示法，这对该领域的发展至关重要。

Now take another example within the region of pure mathematics. Let us proceed to find the rate of increase of the function for any value of its argument. We have not yet really defined what we mean by rate of increase. We will try to grasp its meaning in relation to this particular case. When increases to , the function increases to ; so that the total increase has been , due to an increase in the argument. Hence throughout the interval to the average increase of the function per unit increase of the argument is . But

现在我们来看一个纯数学领域中的另一个例子。我们来求函数 在任意自变量 处的增长率。此时我们还没有真正定义“增长率”的含义，我们将尝试通过这个特定的例子来理解它的意义。

当 增加到 时，函数 增加到 ；因此函数的总增量是 ，这是由于自变量增加了 所导致的。于是，在从 到 这个区间内，函数每单位自变量的平均增量是 。但是 and therefore

并且因此 Thus is the average increase of the function per unit increase in the argument, the average being taken over by the interval to . But depends on , the size of the interval. We shall evidently get what we want, namely the rate of increase at the value of the argument, by diminishing more and more. Hence in the limit when has decreased indefinitely, we say that is the rate of increase of at the value of the argument.

因此， 是函数 在自变量每单位增加时的平均增长量，这个平均值是取在从 到 的区间上的。但 是依赖于 的，也就是依赖于这个区间的长度。显然，我们想要得到的是在自变量为 时的瞬时增长率（也就是“增长率”本身），那么我们就需要让 越来越小。因此，在极限情形下，当 无限趋近于零时，我们就说 是函数 在自变量取值为 时的增长率。

Here again we are apparently driven up against the idea of infinitely small quantities in the use of the words 'in the limit when has decreased indefinitely'. Leibniz held that, mysterious as it may sound, there were actually existing such things as infinitely small quantities, and of course infinitely small numbers corresponding to them. Newton's language and ideas were more on the modern lines; but he did not succeed in explaining the matter with such explicitness so as to be evidently doing more than explain Leibniz's ideas in rather indirect language. The real explanation of the subject was first given by Weierstrass and Berlin School of mathematicians about the middle of the nineteenth century. But between Leibniz and Weierstrass a copious literature, both mathematical and philosophical, had grown up round these mysterious infinitely small quantities which mathematics had discovered and philosophy proceeded to explain.

在这里，我们似乎又一次不得不面对“无穷小量”的概念，因为我们使用了“当 无限减小时的极限”这样的说法。莱布尼茨认为，尽管听起来神秘，实际上确实存在所谓的无穷小量，当然也存在与之对应的无穷小的数。牛顿的语言和思想则更接近现代观点；但他并没有成功地将这个问题解释得足够清晰，以至于他的表述看起来更像是以一种比较间接的方式在阐述莱布尼茨的思想。这个问题的真正解释，最早是由魏尔施特拉斯（Weierstrass）和十九世纪中期柏林学派的数学家们给出的。但在莱布尼茨与魏尔施特拉斯之间，围绕这些神秘的、由数学发现而哲学试图加以解释的“无穷小量”，已经发展出了一大批数学与哲学的著作。

Some philosophers, Bishop Berkeley, for instance, correctly denied the validity of the whole idea, though for reasons other than those indicated here. But the curious fact remained that, despite all criticisms of the foundations of the subject, there could be no doubt but that the mathematical procedure was substantially right. In fact, the subject was right, though the explanations were wrong. It is this possibility of being right, albeit with entirely wrong explanations as to what is being done, that so often makes external criticism—that is so far as it is meant to stop the pursuit of a method—— singularly barren and futile in the progress of science. The instinct of trained observers, and their sense of curiosity, due to the fact that they are obviously getting at something, are far safer guiders. Anyhow the general effect of the success of the Differential Calculus was to generate a large amount of bad philosophy, centring round the idea of the infinitely small. The relics of the verbiage may still be found in the explanations of many elementary mathematical text-books on the Differential Calculus. It is a safe rule to apply that , when a mathematical of philosophical author writes with a misty profundity, he is talking nonsense.

一些哲学家（例如贝克莱主教）确实正确地否定了整个观念的有效性，尽管他们的理由与此处所提出的不同。但奇怪的事实仍然是，尽管对这一学科的基础存在各种批评，却毫无疑问，这一数学操作本质上是正确的。事实上，这一学科本身是对的，尽管对它的解释是错的。

正是这种“操作本身是正确的，而对其所做之事的解释却完全错误”的可能性，常常使得来自外部的批评——尤其是那些意在阻止一种方法继续探索的批评——在科学发展中显得格外空洞而无效。训练有素的观察者的直觉，以及他们那源自“显然发现了某种东西”的好奇心，要比外部批评更值得依赖。

无论如何，微积分的成功在总体上产生了大量糟糕的哲学，其核心围绕着“无穷小”的观念展开。如今在许多关于微积分的初等数学教材中，仍可看到这类冗词赘语的残留。一个值得遵循的安全法则是：当某位数学或哲学作者用朦胧而深奥的语言写作时，他多半是在说废话。

Newton would have phrased the question by saying that, as approaches zero, in the limit become . It is our task so to explain this statement as to show that it does not in reality covertly assume the existence of Leibnize's infinitely small quantities. In reading over the Newtonian method of statement, it is tempting to seek simplicitly by saying that is , when is zero. But this will not do; for it thereby abolishes the interval from to , over which the average increase was calculated. The problem is, how to keep an interval of length over which to calculate the average increase, and at the same time to treat as if it were zero. Newton did this by the conception of a limit, and we now proceed to give Weierstrass's explanation of its real meaning.

牛顿会这样表述这个问题：当 趋近于零时，在极限中， 变为 。我们的任务是这样解释这句话，以便说明它实际上并未暗中假设莱布尼茨所说的无穷小量的存在。在阅读牛顿式的表达方式时，人们往往会因为追求简单而倾向于直接说“当 等于零时， 就是 ”。但这种说法是行不通的，因为它抹除了从 到 的区间——正是在这个区间上我们计算了平均增量。问题在于：我们既要保留一个长度为 的区间，用以计算平均增长率，同时又要把 当作趋近于零来处理。牛顿通过“极限”的概念来实现这一点，而我们接下来将给出魏尔斯特拉斯（Weierstrass）对这一概念真正含义的解释。

In the first place notice that, in discussing , we have been considering as fixed in value and as varying. In other words has been treated as a 'constant' variable, or parameter, as explained in Chapter 9; and we have really been considering as a function of the argument . Hence we can generalize the question on hand, and ask what we mean by saying that the function tends to the limit , say, as its argument tends to the value zero. But again we shall see that the special value for the argument does not belong to the essence of the subject; and again we generalize still further, and ask, what we mean by saying that the function tends to the limit as tend to the value .

首先请注意，在讨论 时，我们一直是将 视为固定不变的，而将 视为变化的。换句话说， 被当作一个“常数变量”或称“参数”（正如第9章所解释的），而我们实际上是在把 当作 的一个函数来研究。

因此，我们可以对当前的问题进行推广，来探讨这样一个问题：当变量 趋近于某个值（比如说零）时，函数 趋近于某个极限值 ，这句话到底是什么意思？

但我们也将看到，参数取值为“零”这一点其实并非这个问题的本质；于是我们可以进一步推广问题：当 趋近于某个值 时，所谓函数 趋近于极限值 ，又是指什么？

Now according to the Weierstrassian explanation the whole idea of tending to the value , though it gives a sort of metaphorical picture of what we are driving at, is really off the point entirely. Indeed it is fairly obvious that, as long as we retain anything like ' tending to ', as a fundamental idea, we are really in the clutches of the infinitely small; for we imply the notion of being infinitely near to . This is just what we want to get rid of.

根据魏尔斯特拉斯（Weierstrass）所提出的解释，所谓“ 趋近于 ”这一整个概念，尽管从比喻的角度提供了一种我们想表达的直观图像，但实际上完全偏离了重点。 事实上，很明显，只要我们还保留“ 趋近于 ”这样的说法作为基本思想，我们就仍然受困于“无穷小”的观念，因为这暗含了“ 无限接近 ”的意思。 而这正是我们想要摆脱的东西。

Accordingly, we shall yet again restate out phrase to be explained, and ask what we mean by saying that the limit of the function at is .

因此，我们将再次重新表述我们要解释的短语，并且提出这样一个问题：当我们说“函数 在 处的极限是 ”时，这到底是什么意思。

The limit of at is a property of the neighbourhood of , where 'neighbourhood' is used in the sense defined in Chapter 11, during the discussion of the continuity of functions. The value of the function at is ; but the limit is distinct in idea from the value, and may be different from it, and may exist when the value has not been defined. We shall also use the term 'standard of approximation' in the sense in which it is defined in Chapter 11. In fact, in the definition of 'continuity' given towards the end of that chapter we have practically defined a limit. The definition of a limit is:

函数 在 处的极限是 的邻域的一个性质，其中“邻域”一词的含义，采用的是第11章中在讨论函数连续性时所定义的含义。函数 在 处的函数值是 ；但极限在概念上不同于函数值，二者可能不同，甚至在函数值尚未定义时，极限仍然可能存在。

我们也将使用“逼近标准”这一术语，其含义同样参考第11章中的定义。事实上，在该章末尾所给出的“连续性”定义中，我们实际上已经定义了极限。极限的定义如下：

A function has the limit at a value of its argument , when in the neighbourhood of its values approximate to within standard of approximation.

当函数 在其自变量 的某个取值 的邻域内，其函数值在任意逼近标准之内趋近于 ，我们就说函数 在 处的极限是 。

Compare this definition with that already given for continuity, namely:

将这个定义与之前给出的连续性的定义进行比较，即：

A function is continuous at a value of its argument, when in the neighbourhood of its values approximate to its value at within standard of approximation.

当函数 在其自变量 的某个取值 处连续时，意味着在 的邻域内，函数的值在任意逼近标准内趋近于函数在 处的值。

IMG_3555

It is at once evident that a function is continues at when (1) it possesses a limit at , and (2) that limit is equal to its value at . Thus the illustrations of continuity which have been given at the end of Chapter 11 are illustrations of the idea of a limit, namely, they were all directed to proving that was the limit of at for the functions considered and the value of considered. It is really more instructive to consider the limit at a point where a function is not continuous. For example, consider the function of which the graph is give in Fig.20 of Chapter 11. This function is defined to have the value 1 for all values of the argument except the integers and for these integral values it has the value . Now let us think of its limit when . We notice that in the definition of the limit the value of the function at (in this case , ) is excluded. But, excluding , the values of , when lies within any interval which (1) contains not as an end-point, and (2) does not extend so far as and , are all equal to ; and hence these values approximate to within every standard of approximation. Hence is the limit of at the value of the argument , but by definition .

显然，当函数在 处连续时，(1) 它在 处有极限，且 (2) 该极限等于函数在 处的值。因此，第11章结尾给出的连续性的例子，实际上是极限的例子，换句话说，它们的目的是证明在所考虑的函数和 处， 是 在 处的极限。实际上，更有启发性的是考虑一个函数在不连续点的极限。例如，考虑第11章图20中给出的函数图像。这个函数 定义为：对于除了整数 以外的所有自变量取值，其函数值为 ；而对于这些整数值，函数值为 。现在让我们思考当 时的极限。我们注意到，在极限的定义中，函数在 处的值（在此例中为 ）是被排除在外的。但是，排除掉 后， 的值，当 落在一个区间内时，满足：(1) 该区间包含 ，但 不是端点；(2) 该区间不会扩展到 和 之间。这些值都等于 ；因此，这些值在任何逼近标准下都趋近于 。因此， 是 在 处的极限，但根据定义，。

This is an instance of a function which possesses both a value and a limit at the value of the argument, but the value is not equal to the limit. At the end of Chapter 11 the function was considered at the value 2 of the argument. Its value at is , i.e. , and it was proved that its limit is also . Thus here we have a function with a value and a limit which are equal.

这是一个例子，说明一个函数在自变量值为 时既有函数值又有极限，但函数值不等于极限。在第11章结尾，讨论了函数 在自变量取值为 2 时的情况。其在 处的函数值为 ，即 ，并且已证明其极限也是 。因此，在这个例子中，我们有一个函数，其函数值和极限是相等的。

Finally we come to the case which is essentially important for our purposes, namely, to a function which possesses a limit, but no defined value at a certain value of its argument. We need not go far to look for such a function, will serve our purpose. Now in any mathematical book, we might find the equation, , written without hesitation or comment. But there is a difficulty in this; for when is zero, ; and has no defined meaning. Thus the value of the function at has no defined meaning. But for every other value of , the value of the function is . Thus the limit of at is , and it has not value at . Similarly the limit of at is whatever may be, so that the limit of at is . But the value of at takes the form , which has no defined meaning. Thus the function has a limit but no value at .

最后，我们来讨论一个对我们目的至关重要的情况，即一个具有极限但在某个自变量值处没有定义值的函数。我们不需要远行就能找到这样的函数， 就能满足我们的要求。现在，在任何数学书籍中，我们可能会看到方程 被毫不犹豫地写出，而没有任何评论。但这里存在一个问题；因为当 为零时，；而 没有定义的意义。因此，函数 在 处没有定义的值。但对于其他所有 的值，函数 的值为 。因此， 在 处的极限是 ，但它在 处没有值。同样地， 在 处的极限是 ，无论 是什么，所以 在 处的极限是 。但是， 在 处的值为 ，而 没有定义的意义。因此，函数 在 处有极限，但没有值。

We now come back to the problem from which we started this discussion on the nature of a limit. How are we going to define the rate of increase of the function at any value of its argument? Our answer is that this rate of increase is the limit of the function at the value zero for its argument . (Note that is here a 'constant'.) Let us see how this answer works in the light of our definition of a limit. We have

我们现在回到我们开始讨论极限性质时的问题。我们如何定义函数 在其自变量的任意值 处的增速呢？我们的答案是，这个增速是函数 在其自变量 取值为零时的极限。（注意， 在这里是一个“常数”。）让我们看看这个答案如何根据我们对极限的定义来解释。 Now in finding the limit of at the value of the argument , the value (if any) of the function at is excluded. But for all values of , except , we can divide through by . Thus the limit of at is the same as that of at . Now, whatever standard of approximation we choose to take, by considering the interval from to we see that, for values of which fall within it, differs from by less than , that is by less than . This is true for standard . Hence in the neighbourhood of the value for , approximates to within standard of approximation, and therefore is the limit of at . Hence by what has been said above is the limit of at the value of for . It follows, therefore, that is what we have called the rate of increase of at the value of the argument. Thus this method conducts us to the same rate of increase for as did the Leibnizian way of making grow 'infinitely small'.

现在，在求解 在 处的极限时，函数在 处的值（如果有的话）被排除在外。但是对于除了 以外的所有 值，我们可以将整个表达式除以 。因此， 在 处的极限与 在 处的极限相同。现在，无论我们选择什么标准的近似 ，通过考虑区间从 到 ，我们看到，对于落在该区间内的 值， 与 的差异小于 ，即小于 。这对于任何标准的 都成立。因此，在 取值为 的邻域内， 以每个标准的近似度趋近于 ，因此 是 在 处的极限。因此，根据上述说法， 是 在 处的极限。因此， 是我们所称的 在自变量值 处的增速。这样，这种方法得出的 的增速与 Leibniz 的方法得到的结果相同，即让 变得“无限小”。

The more abstract terms 'differential coefficient', or 'derived function', are generally used for what we have hitherto called the 'rate of increase' of a function. The general definition is as follows, the differential coefficient of the function is the limit, if it exist, of the function of the argument at the value of its argument.

更抽象的术语“微分系数”或“导函数”通常用来表示我们此前所称的“函数的增速”。一般的定义如下：函数 的微分系数是函数 在其自变量 取值为 时的极限（如果存在的话）。

How have we, by this definition and the subsidiary definition of a limit, really managed to avoid the notion of 'infinitely small numbers' which so worried our mathematical forefathers? For them the difficulty arose because on the one hand they had to use an interval to over which to calculate the average increase, and, on the other hand, they finally wanted to put . The result was they seemed to be landed into the notion of an existent interval of zero size. Now how do we avoid this difficulty? In this way— we use the notion that corresponding to standard of approximation, interval with such and such properties can be found. The difference is that we have grasped the importance of the notion of 'the variable', and they had not done so. Thus , at the end of our exposition of the essential notions of mathematical analysis, we are led back to the ideas with which in Chapter 2 we commenced our inquiry— that in mathematics the fundamentally important ideas are those of 'some things' and 'any things'.

通过这个定义以及附带的极限定义，我们是如何真正避免了“无穷小量”这一曾困扰我们数学先驱者的概念呢？他们面临的困难在于，一方面他们必须使用从 到 的区间来计算平均增速，而另一方面，他们最终又想让 。结果似乎让他们陷入了一个存在零大小区间的概念。那么我们是如何避免这一困难的呢？我们是通过这种方式——我们使用了这样一个概念：对于任意标准的逼近，某个具备特定属性的区间是可以找到的。区别在于，我们已经理解了“变量”这一概念的重要性，而他们并没有做到这一点。因此，在我们对数学分析基本概念的阐述结束时，我们回到了在第二章开始我们研究时所用的思想——在数学中，根本重要的概念是“某些事物”和“任何事物”的概念。